A329979 Prime numbers which can be represented as p^i * q^j - (p + q) where p and q are distinct odd primes and i,j > 0.

7, 11, 19, 23, 31, 37, 43, 47, 53, 59, 67, 71, 79, 83, 101, 103, 107, 127, 131, 137, 139, 149, 163, 167, 179, 181, 191, 199, 211, 223, 229, 233, 239, 251, 263, 271, 283, 293, 307, 311, 331, 347, 349, 359, 367, 373, 379, 383, 397, 419, 421, 431, 439, 443, 463, 467, 479, 491, 499

Offset: 1

Author

Craig J. Beisel, Nov 26 2019

Keywords

• nonn
• changed

Comments

Numbers of this form are an attempt to generalize Mersenne numbers (see link).

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Craig J. Beisel, Can it be shown that numbers of a certain form produce primes more often than expected?, Math StackExchange, November 2019.

Programs

Maple

N:= 1000: # for terms <= N
P:= select(isprime, [seq(i,i=3..(N+3)/2,2)]):
S:= {}:
for ip from 1 to nops(P) do
  p:= P[ip];
  for i from 1 while p^i*3 - (p+3) <= N do
    for iq from 1 to ip-1 do
       q:= P[iq];
       if p^i*q - (p+q) > N then break fi;
       for j from 1 do
         x:= p^i * q^j - (p+q);
         if x > N then break fi;
         if isprime(x) then S:= S union {x} fi;
od od od od:
sort(convert(S,list));  # Robert Israel, Aug 25 2025

PARI

z=[];forprime(a=3,1000, forprime(b=a+2,1000, for(i=1,10, for(j=1,10, y=a+b; x=a^i*b^j-y; if(x<500 && isprime(x) && setsearch(z,x)==0,z=setunion(z,[x])) )))); print(z)

Extensions

Definition clarified by Robert Israel, Aug 25 2025